Stacking of oligo and polythiophenes cations in solution: surface tension and dielectric saturation

The stacking of positively charged (or doped) terthiophene oligomers and quaterthiophene polymers in solution is investigated applying a recently developed unified electrostatic and cavitation model for first-principles calculations in a continuum solvent. The thermodynamic and structural patterns of the dimerization are explored in different solvents, and the distinctive roles of polarity and surface tension are characterized and analyzed. Interestingly, we discover a saturation in the stabilization effect of the dielectric screening that takes place at rather small values of $\epsilon_0$. Moreover, we address the interactions in trimers of terthiophene cations, with the aim of generalizing the results obtained for the dimers to the case of higher-order stacks and nanoaggregates.


I. INTRODUCTION
The intermolecular interactions between conjugated polymers and oligomers in the condensed phase-whether in the solid state or in solution-entail a fundamental interest in the emerging field of molecular electronics, as they are decisive factors in the electronic and structural properties of these materials. The orientation and alignment of polymers in a solid matrix, the formation of aggregates in films and solution, or the ability of organic semiconductors to self-assemble are the outcome of a complex balance between the spatial features of the molecules and the substrate, and the interactions between them at the conditions of synthesis. 1 −6 By dictating rules for aggregation, these interactions eventually shape properties such as charge delocalization and mobility 2,6−8 or optical 3,9−11 and electromechanical 12 response.
Thiophene-derived oligomers and polymers represent today one of the most promising class of organic semiconductors, finding potential applications in a variety of electronic and electroactive devices. 13 −15 Semiconducting properties arise with doping, therefore much of the basic research performed on these systems has addressed in particular the doped or oxidized species. Since the early nineties electrochemical and spectroscopic evidence was gathered indicating that oxidized oligothiophenes reversibly associate in solution. 16−18 In a recent study, 19 we have outlined how this association is driven by three contributions: the attractive π-π interactions, the Coulombic repulsion, and the solvent effects. In the case of oligothiophene cations dimers, combination of semioccupied HOMOs form occupied bonding and empty antibonding orbitals, resulting in an interaction of covalent character, different in nature to the one arising in neutral dimers, of dispersive origin. 20 In vacuum, the electrostatic repulsion between the cations largely exceeds the covalent term, 19,21 making apparent the importance of the solvent (or of the counterions in the solid state case) in stabilizing the stacks. A polarizable dielectric medium favors concentration of charge in a small cavity, reverting the balance from net repulsion to attraction and stacking.
In the present paper, we employ a recently developed first-principles approach recently developed to describe the effect of a continuum solvent within the density-functional theory framework, 22 and use it to explore the role of polarity and surface tension in the stabilization of dimers of polythiophene and oligothiophene radical cations. Furthermore, we examine the possibility of trimer formation as an intermediate step toward the nucleation of higher-order aggregates, and to gain insight on self-assembly in solution. In charged dimers, at variance with the case of neutral dimers governed by van-der-Waals forces, interactions are predominantly covalent and electrostatic, and density-functional theory (DFT) has proven to be sufficiently accurate when compared with highly-correlated quantum-chemistry methods. 19 In our approach, the contribution of the surface tension to the solvation free energy is computed in a very natural fashion, as the product between the area of the cavity and the surface tension of the solvent. 22 This contribution is particularly important in dimerization processes, where the merging of two cavities into one provides an additional stabilizing term associated to the minimization of the total area of the cavity. Solvation is described with a continuum model recently implemented by us in the Quantum-ESPRESSO package and described in detail in reference 22 . In this approach, the solvent is represented as a dielectric medium surrounding a quantum-mechanical solute confined in a cavity delimited by an isosurface of electronic charge density. Adopting a common decomposition of the solvation free energy ∆G sol we have: where ∆G el , ∆G cav , and ∆G dis−rep are the electrostatic, the cavitation, and the dispersionrepulsion contributions respectively. 26 In our implementation ∆G el and ∆G cav are considered explicitly, while ∆G dis−rep is largely captured (by virtue of the parametrization) by the electrostatic term. In the following, we briefly review the approaches used to obtain ∆G el and ∆G cav .
The electrostatic interaction between the dielectric medium and the solute is calculated, as proposed by Fattebert and Gygi, 27,28 by solving the Poisson equation in the presence of a dielectric continuum with permittivity ǫ[ρ]: The function ǫ[ρ] is defined to asymptotically approach the permittivity of the bulk solvent ǫ 0 in regions of space where the electron density is low, and to approach 1 in those regions where it is high. 22 In this way the dielectric medium and the electronic density respond selfconsistently to each other through the dependence of ǫ on ρ and vice-versa. The variation in the dielectric constant at the solvent-solute interface is controlled by two parameters ρ 0 and β, which determine the size of the cavity and the smoothness of the transition region.
These are the only parameters entering the model, and our chosen values, ρ 0 =0.00078 e and β=1.3, represent a rather universal choice. 22 The cavitation term is computed as the product between the surface tension of the solvent γ and the area of the cavity, where S(ρ 0 ) is the surface of the same cavity employed in the electrostatic part of the solvation energy and is defined by an isosurface of the charge density. This area can be easily and accurately calculated by integration in a real-space grid, as the volume of a thin film delimited between two charge density isosurfaces, divided by the thickness of this film.
This idea has been originally proposed by Cococcioni et al. 29 to define a "quantum surface" in the context of extended electronic-enthalpy functionals: The finite-differences parameter ∆ determines the separation between two adjacent isosurfaces, one external and one internal, corresponding to density thresholds ρ 0 − ∆/2 and ρ 0 + ∆/2 respectively. The spatial distance between these two cavities-or the thickness of the film-is given at any point in space by the ratio ∆/|∇ρ|. The (smoothed) step function ϑ is zero in regions of low electron density and approaches 1 otherwise, and it has been defined consistently with the dielectric function ǫ[ρ].

III. RESULTS AND DISCUSSION
Crystallographic data 30  A local minimum at 0.0Å (where the two layers are overlapping) is present in both cases.
Interestingly, the net binding is very sensitive to the lateral shift, varying steeply in a range of 10 kcal/mol as one layer is slipped over the other. At shifts of about 1Å off the minima, the π-π interaction between the cations appears clearly weakened, resulting in an unbound dimer.
The nature of the solvent doesn't have any significant effect on this characteristic pattern, even if it affects the magnitude of the interaction. This is shown in Fig. 3, where the terthiophene curve is displayed for three different media: acetonitrile, dichloromethane showing that ∆H d is enhanced by the length of the chain, 17 a trend related to a "dilution" of the Coulombic repulsion as the ratio between charge and oligomer size decreases. 19 At the same time, however, the increase in length at a given oxidation state would diminish the ratio between unpaired electrons available to π-π bonding and thiophene rings, what would presumably revert the aforementioned binding trend starting from certain molecular weighs. 35 The separate roles played by the dielectric screening of the solvent and its surface tension in the stabilization of the dimer are highlighted in Fig. 6. If the contribution of ∆G cav were omitted, the binding curves would turn out to be very close to each other (Fig. 6a). The  Fig. 6a respectively) overlap on the right part of the plot.
The potential energy curves in Fig. 6a unveil an intriguing possibility: that the binding energy is not directly related to the dielectric constant of the solvent, as our intuition may suggest. This hypothesis is explored in Fig. 7, where the interaction energy between two terthiophene cations separated by 3.6Å is plotted as a function of the dielectric constant, ignoring the contribution of the surface tension. The results are somehow unexpected: a rather small increase in the permittivity with respect to the vacuum limit rapidly stabilizes the dimer, but once the dielectric constant is above 10 the effect of a further increase in polarity is very small. This behavior can be rationalized considering that a polarizable dielectric medium with low permittivity is already enough to screen most of the Coulombic repulsion between the two charges and to favor aggregation of these charges by polarizing itself. We note in passing that the positive drift observed at higher permittivities for the case of ρ 0 =0.00078 e is an artifact of the continuum model. Since the dielectric constant is defined as a continuous function of the electron density, its value throughout the intradimer region may depart from 1, allowing the dielectric medium to fill some of the space between the cations and to interfere, though modestly, with the π-π bond. This effect will be enhanced at large values of ǫ 0 and ρ 0 . In reality, instead, the solvent does not penetrate the intradimer space if the separation is 3.6Å, regardless of ǫ 0 . This spurious behavior is in fact absent in the curve computed with ρ 0 =0.0003 in Fig. 7. What is remarkably captured by the continuum model is the saturation effects of polarity on the dimerization, occurring already for very low dielectric constants. These results are pretty much consistent with experimental observations that turn down a direct correlation between dimerization trends and polarity of the medium, while emphasizing the dependence on solubility of the oligothiophenes. 36 To understand the effect of the solvent on ∆H d , then, one should consider other properties such as surface tension or specific interactions between the solute and the medium.
Are the thermodynamic and structural features found so far for the dimerization applicable to the stacking of multiple oligomer layers? It would be very interesting to know if or how the present results can be extended to processes such as aggregation and self-assembly in solution, involving the collective pairing of many oligothiophene units. In an attempt to offer an answer, even if preliminary, to this question, we have studied the formation of trimers of terthiophene cations in acetonitrile. Fig. 8 depicts the two configurations of minimum energy obtained for the trimer in acetonitrile, in which the third cation is shifted by + or -2.3Å with respect to the next oligomer. As shown in Fig. 9, where the interaction energy is plotted as a function of the lateral shift of the third cation, there is no significant energetic difference between these two minima. The curve corresponding to the dimer is plotted in the same figure: the pattern of valleys and peaks is preserved at the same lateral displacements when increasing the number of layers from two to three. The differences in the relative depths of these curves can be ascribed to the fact that the same interplanar separation of 3.4Å was adopted in the calculation of both, but the optimal separation in the trimer is longer, as can be seen in Fig. 10. This graph shows the interaction energy calculated for the trimer in acetonitrile as a function of the interplanar separation between layers (the interplanar separation between the first and the second layer is the same as between the second and the third at each point of the curve). For meaningful comparison with the dimer, depicted in the same graph, the energies were normalized to the number of π-pairs, in this case two. Interestingly, the binding between two cations doesn't seem to be impaired by the presence of a third one: the interaction between stacks remains almost constant, even though the equilibrium distance increases in about 0.1Å. This suggests that the energetic and structural results found for the cation dimers can be applied, to a large extent, to the case of more complex, larger aggregates consisting of multiple layers.

IV. SUMMARY
Our study has highlighted the separate roles of surface tension the dielectric screening in the stabilization of charged thiophene oligomers and polymers stacks. The surface tension of the solvent is a driving force toward the minimization of the cavity area, and therefore toward dimerization: there is an energetic payoff in accommodating two solutes in a single cavity of an area smaller than twice the one corresponding to the dissociated components.
On the other hand, the dependence of the dimer stability on the polarity of the solvent alone is less evident. A dielectric effect is necessary to screen the electrostatic repulsion and to stabilize the charges in a small volume, but once the permittivity has reached a certain threshold, a further increase in polarity has a negligible contribution to the stabilization of the system. This observation is probably general to any π-dimer of charged radicals-an hypothesis that could be interesting to test through explicit calculation.
The formation of trimers follows the same geometrical arrangement as the dimerization.
A π-bond on one of the oligomer planes does not seem to significantly affect the bond on the other. These results point to the conclusion that the organization of aggregates and stacks is governed by the same thermodynamics that is already manifest in the dimerization.